Math Formulas/Equations

What You Need to Know

On SAT Math, a huge share of points comes from recognizing which equation/formula models the situation and then manipulating it cleanly. This topic is about your core toolbox:

• Algebra rules (distributing, factoring, exponents/radicals)

• Solving equations/inequalities (linear, quadratic, rational, absolute value)

• Core function models (linear, exponential)

• Coordinate geometry equations (line formulas, circle equation)

If you can (1) rewrite expressions correctly and (2) solve for the variable without illegal moves, you’ll crush most “equation” questions.

Core idea

An equation states two expressions are equal and you’re finding values that make it true. An inequality compares expressions with $, \ge$ and you’re finding values that make it true.

Step-by-Step Breakdown

A) Solving linear equations (one variable)

1. Simplify each side: distribute, combine like terms.

2. Get variables on one side (add/subtract terms).

3. Isolate the variable (multiply/divide).

4. Check for special cases:

   • No solution: variable cancels and you get false, e.g. $0 = 5$.

   • Infinitely many solutions: you get true, e.g. $0 = 0$.

Mini-example: Solve $3(x-2)=2x+5$

• Distribute: $3x-6=2x+5$

• Subtract $2x$: $x-6=5$

• Add $6$: $x=11$

B) Solving linear inequalities

1. Solve like an equation.

2. Flip the inequality only if you multiply/divide by a negative.

3. Write in interval form if asked.

Mini-example: Solve $-2x+1 \le 7$

• Subtract $1$: $-2x \le 6$

• Divide by $-2$ (flip!): $x \ge -3$

C) Solving systems of linear equations

Use substitution or elimination.

Elimination steps:

1. Align equations as $ax+by=c$ format.

2. Multiply one/both equations so a variable’s coefficients are opposites.

3. Add/subtract to eliminate.

4. Solve for remaining variable.

5. Back-substitute.

Mini-example:
Solve
$x+y=9$
$x-y=1$
Add: $2x=10 \Rightarrow x=5$, then $y=4$.

D) Solving quadratics

Quadratics appear as $ax^2+bx+c=0$.

Method choice (fast decision):

1. If it factors nicely, factor.

2. If it’s already $a(x-h)^2+k$, use vertex form.

3. If not factorable, use the quadratic formula.

Factoring steps:

1. Move everything to one side: $=0$.

2. Factor.

3. Set each factor to zero.

Mini-example: $x^2-5x+6=0$
Factor: $(x-2)(x-3)=0$ so $x=2$ or $x=3$.

E) Rational equations (variables in denominators)

1. State restrictions: denominator $\ne 0$.

2. Multiply both sides by the LCD (least common denominator).

3. Solve the resulting equation.

4. Check for extraneous solutions (plug back).

Mini-example: Solve $\frac{2}{x-1}=3$

• Restriction: $x \ne 1$

• Multiply: $2=3(x-1)=3x-3$

• $3x=5 \Rightarrow x=\frac{5}{3}$ (valid)

F) Absolute value equations/inequalities

Equation rule:
$|A|=b$ (with $b\ge 0$) becomes $A=b$ **or** $A=-b$.

Inequality rules:

• $|A|</p></li><li><p>$|A|>b$becomes$A>b$**or**$A<-b$</p></li></ul><p><strong>Mini-example:</strong>$|2x-1|=5$</p><ul><li><p>$2x-1=5 \Rightarrow x=3$</p></li><li><p>$2x-1=-5 \Rightarrow x=-2$</p></li></ul><h5 id="cb301cd8-5dd7-4fa4-a08f-6b156b747c6f" data-toc-id="cb301cd8-5dd7-4fa4-a08f-6b156b747c6f" collapsed="false" seolevelmigrated="true">G) Exponential equations (SAT-level)</h5><p>Often solvable by rewriting to a common base.</p><p><strong>Mini-example:</strong> Solve$2^{x+1}=8$<br>Rewrite$8=2^3$, so$x+1=3 \Rightarrow x=2.

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  Key Formulas, Rules & Facts

  Algebra + equation-solving essentials

  Formula / Rule	When to use	Notes / pitfalls
  a(b+c)=ab+ac$</p></td><td colspan="1" rowspan="1"><p>Distribute</p></td><td colspan="1" rowspan="1"><p>Don’t forget to distribute to <strong>every</strong> term</p></td></tr><tr><td colspan="1" rowspan="1"><p>$ab+ac=a(b+c)$</p></td><td colspan="1" rowspan="1"><p>Factor (GCF)</p></td><td colspan="1" rowspan="1"><p>Always check for a <strong>GCF</strong> first</p></td></tr><tr><td colspan="1" rowspan="1"><p>$(x+m)(x+n)=x^2+(m+n)x+mn$</p></td><td colspan="1" rowspan="1"><p>Multiply binomials</p></td><td colspan="1" rowspan="1"><p>Useful to reverse when factoring</p></td></tr><tr><td colspan="1" rowspan="1"><p>$x^2-y^2=(x-y)(x+y)$</p></td><td colspan="1" rowspan="1"><p>Difference of squares</p></td><td colspan="1" rowspan="1"><p>Common fast factor on SAT</p></td></tr><tr><td colspan="1" rowspan="1"><p>$(x\pm y)^2=x^2\pm 2xy+y^2$</p></td><td colspan="1" rowspan="1"><p>Perfect square trinomials</p></td><td colspan="1" rowspan="1"><p>Spot patterns for fast factoring</p></td></tr><tr><td colspan="1" rowspan="1"><p>If$ab=0$, then$a=0$or$b=0$</p></td><td colspan="1" rowspan="1"><p>Zero product property</p></td><td colspan="1" rowspan="1"><p>Only works when product equals <strong>0</strong></p></td></tr><tr><td colspan="1" rowspan="1"><p>$\frac{a}{b}=\frac{c}{d} \Rightarrow ad=bc$</p></td><td colspan="1" rowspan="1"><p>Proportions</p></td><td colspan="1" rowspan="1"><p>Ensure$b\ne 0$and$d\ne 0

  Exponents & radicals

  Rule	When to use	Notes / pitfalls
  a^m\cdot a^n=a^{m+n}$</p></td><td colspan="1" rowspan="1"><p>Multiply same base</p></td><td colspan="1" rowspan="1"><p>Bases must match</p></td></tr><tr><td colspan="1" rowspan="1"><p>$\frac{a^m}{a^n}=a^{m-n}$</p></td><td colspan="1" rowspan="1"><p>Divide same base</p></td><td colspan="1" rowspan="1"><p>If$m
  $(a^m)^n=a^{mn}$	Power of a power	Common place to slip
  $(ab)^n=a^n b^n$	Distribute exponent	Works for multiplication
  $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$	Power of a fraction	Need $b\ne 0$
  $a^{-n}=\frac{1}{a^n}$	Negative exponents	Means “reciprocal”
  $\sqrt{a}\sqrt{b}=\sqrt{ab}$ (for $a,b\ge 0$)	Multiply radicals	Don’t combine across plus: $\sqrt{a}+\sqrt{b}$ stays
  $\sqrt{a^2}=|a|$	Simplifying radicals	SAT may test absolute value nuance

  Linear functions & lines

  Formula	When to use	Notes
  $m=\frac{y_2-y_1}{x_2-x_1}$	Slope between two points	Vertical line: $x_2=x_1$ (undefined slope)
  $y=mx+b$	Line form	$b$ is $y$-intercept
  $y-y_1=m(x-x_1)$	Point-slope form	Great when you have slope + a point
  Standard form $Ax+By=C$	Systems / intercepts	Slope is $-\frac{A}{B}$ (if $B\ne 0$)
  Parallel lines: $m_1=m_2$	Identify parallel	Same slope
  Perpendicular: $m_1m_2=-1$	Identify perpendicular	Negative reciprocal slopes

  Coordinate geometry

  Formula	When to use	Notes
  Distance: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$	Segment length	Comes from Pythagorean theorem
  Midpoint: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$	Midpoint of segment	Average the coordinates
  Circle: $(x-h)^2+(y-k)^2=r^2$	Circle graph/equation	Center $(h,k)$, radius $r$

  Quadratics (must-know relationships)

  Formula / fact	When to use	Notes
  Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$	Solve any quadratic	Watch sign of $b$ carefully
  Discriminant: $\Delta=b^2-4ac$	Number of real solutions	\Delta>0 two, $\Delta=0$ one, \Delta<0 none (real)
  Vertex: for $ax^2+bx+c$, $x=\frac{-b}{2a}$	Find axis of symmetry	Then plug in for $y$
  Vertex form: $a(x-h)^2+k$	Graph transformations	Vertex at $(h,k)$

  Percent, rate, and growth equations

  Equation	When to use	Notes
  Percent change: $\frac{\text{new}-\text{old}}{\text{old}}\times 100\%$	Increase/decrease	“Of” usually means multiply
  Simple interest: $I=Prt$	Interest problems	$r$ as decimal, $t$ in years
  Exponential growth/decay: $A=A_0(1\pm r)^t$	Repeated percent change	Use -$ for decay, +$ for growth
  Average speed: $v=\frac{d}{t}$	Motion problems	Total avg speed is $\frac{\text{total }d}{\text{total }t}$

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  Examples & Applications

  1) Rearranging a formula (literal equations)

  Problem style: Solve for a variable in a given formula.

  Given $V=\frac{1}{3}\pi r^2 h$, solve for $h$.

  • Multiply both sides by $3$: $3V=\pi r^2 h$

  • Divide by $\pi r^2$: $h=\frac{3V}{\pi r^2}$

  Key insight: Treat it like isolating $x$—just keep operations balanced.

  2) System from a word problem

  A theater sold 100 tickets. Adult tickets cost $\$12$ and student tickets cost $\$8$. Total revenue was $\$1040$. How many student tickets?

  Let $a$ = adult, $s$ = student.

  • Count: $a+s=100$

  • Revenue: $12a+8s=1040$

  Eliminate: multiply first equation by $8$:
  $8a+8s=800$
  Subtract from revenue equation:
  $(12a+8s)-(8a+8s)=1040-800 \Rightarrow 4a=240 \Rightarrow a=60$
  So $s=40$.

  Key insight: “Total number” + “total value” is almost always a 2-equation system.

  3) Quadratic from geometry (area)

  A rectangle has area $48$ and length $x+2$ and width $x-2$. Find $x$.

  Set up: $(x+2)(x-2)=48$
  Use difference of squares: $x^2-4=48$
  So $x^2=52 \Rightarrow x=\pm\sqrt{52}=\pm 2\sqrt{13}$.

  SAT reality check: If $x$ is a dimension parameter, you may need **positive only** (and also ensure x-2>0). So $x=2\sqrt{13}$ works.

  4) Inequality + interval solution

  Solve and graph: 2(1-x)>x+4

  • Distribute: 2-2x>x+4

  • Subtract $2$: -2x>x+2

  • Subtract $x$: -3x>2

  • Divide by $-3$ (flip): x< -\frac{2}{3}

  Key insight: The only “special move” is flipping the sign when dividing by a negative.

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  Common Mistakes & Traps

  1. Forgetting to distribute a negative

     • Wrong: $-(x-3)=-x-3$

     • Right: $-(x-3)=-x+3$

     • Fix: Treat $-1$ as the multiplier and distribute carefully.

  2. Combining unlike terms

     • Wrong: $2x+3=5x$

     • Why wrong: $3$ isn’t an $x$-term.

     • Fix: Only combine terms with the exact same variable part.

  3. Illegal canceling across addition

     • Wrong: $\frac{x+2}{x}=\frac{2}{1}$

     • Why wrong: You can only cancel factors, not terms in a sum.

     • Fix: Factor first if possible.

  4. Not flipping the inequality when dividing by a negative

     • Wrong: -2x<6 \Rightarrow x< -3

     • Right: $-2x -3$

     • Fix: Circle the sign whenever you divide/multiply by a negative.

  5. Dropping solutions when solving quadratics

     • Wrong: Taking only the $+$ in $\pm$ or missing the second factor.

     • Fix: For $x^2=9$, write $x=\pm 3$ unless context restricts.

  6. Extraneous solutions in rational/absolute value equations

     • What happens: Multiplying by a variable expression can introduce invalid answers.

     • Fix: State restrictions (like $x\ne 1$) and plug solutions back.

  7. Misreading function notation

     • Mistake: Thinking $f(x+2)$ equals $f(x)+2$.

     • Fix: Replace the entire input: if $f(x)=x^2$, then $f(x+2)=(x+2)^2$.

  8. Sign errors in the quadratic formula

     • Common slip: Using $-b$ incorrectly when $b$ is negative.

     • Fix: If $b=-5$, then $-b=5$. Put parentheses: $-(-5)=5$.

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  Memory Aids & Quick Tricks

  Trick / mnemonic	What it helps you remember	When to use
  “Same base, add/subtract exponents”	$a^m a^n=a^{m+n}$ and $a^m/a^n=a^{m-n}$	Simplifying exponent expressions
  “SOH-CAH-TOA”	$\sin=\frac{\text{opp}}{\text{hyp}}$ etc.	Only if trig appears (rare), but can help with right-triangle ratios
  “FOIL”	Multiply $(a+b)(c+d)$	Expanding binomials (though distributing is safer)
  “Flip when negative”	Inequality sign flips when dividing/multiplying by negative	Inequalities
  “Circle form = Center/Radius”	$(x-h)^2+(y-k)^2=r^2$ gives center $(h,k)$	Circle equation questions
  “Axis is $-b/2a$”	Quick vertex $x$-coordinate	Quadratic graphs/vertex/maximum-minimum

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  Quick Review Checklist

  • You can distribute, combine like terms, and factor (GCF, difference of squares, perfect square patterns).

  • You know exponent rules, including $a^{-n}=\frac{1}{a^n}$ and $\sqrt{a^2}=|a|$.

  • When solving equations, you isolate the variable and watch for no solution vs infinite solutions.

  • For inequalities, you flip the sign when multiplying/dividing by a negative.

  • For systems, you can do elimination quickly and interpret solutions.

  • For quadratics, you can factor or use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and use $\Delta=b^2-4ac$ to predict roots.

  • For rational equations, you state restrictions and check for extraneous answers.

  • You can use line and coordinate formulas: $m=\frac{y_2-y_1}{x_2-x_1}$, $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$, midpoint, and $(x-h)^2+(y-k)^2=r^2$.

  You don’t need new tricks—just clean algebra and careful sign handling.